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 Description: A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)
Assertion
Ref Expression
trssord ((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → Ord 𝐴)

Proof of Theorem trssord
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dford3 4104 . . . . . . 7 (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥𝐵 Tr 𝑥))
21simprbi 260 . . . . . 6 (Ord 𝐵 → ∀𝑥𝐵 Tr 𝑥)
3 ssralv 3004 . . . . . 6 (𝐴𝐵 → (∀𝑥𝐵 Tr 𝑥 → ∀𝑥𝐴 Tr 𝑥))
42, 3syl5 28 . . . . 5 (𝐴𝐵 → (Ord 𝐵 → ∀𝑥𝐴 Tr 𝑥))
54imp 115 . . . 4 ((𝐴𝐵 ∧ Ord 𝐵) → ∀𝑥𝐴 Tr 𝑥)
65anim2i 324 . . 3 ((Tr 𝐴 ∧ (𝐴𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
763impb 1100 . 2 ((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
8 dford3 4104 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
97, 8sylibr 137 1 ((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → Ord 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 885  ∀wral 2306   ⊆ wss 2917  Tr wtr 3854  Ord word 4099 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-in 2924  df-ss 2931  df-iord 4103 This theorem is referenced by:  ordelord  4118  ordin  4122  ssorduni  4213  ordtriexmidlem  4245  ordtri2or2exmidlem  4251  onsucelsucexmidlem  4254  ordsuc  4287
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